SECTION 8.2 The Law of Sines 563 The third side c can now be determined using the Law of Sines. = ° = ° = ° ° ≈ A a C c c c sin sin sin40 3 sin114.6 3 sin114.6 sin40 4.24 Figure 29(b) illustrates the solved triangle. Figure 29(b) 408 2 3 c 5 4.24 C 5 114.68 B 5 25.48 Using the Law of Sines to Solve an SSA Triangle (Two Solutions) Solve the triangle: = = = ° a b A 6, 8, 35 EXAMPLE 5 Figure 30(b) C1 5 95.18 C2 5 14.98 B1 5 49.98 B2 5 130.18 c2 5 2.69 c1 5 10.42 358 8 6 6 Figure 30(a) B1 5 49.98 B2 5 130.18 358 8 6 6 Solution Because = = a b 6, 8, and = ° A 35 are known, use the Law of Sines to find the angle B. = A a B b sin sin Then ° = = ° ≈ B B sin35 6 sin 8 sin 8 sin35 6 0.76 ≈ ° ≈ °− ° = ° B B 49.9 or 180 49.9 130.1 1 2 Both values of B result in + < ° A B 180 . There are two triangles, one containing the angle ≈ ° B 49.9 1 and the other containing the angle ≈ ° B 130.1 . 2 See Figure 30(a). The third angle C is either = °− − ≈ ° = °− − ≈ ° C A B C A B 180 95.1 or 180 14.9 1 1 2 2 ↑ ↑ = ° = ° A B 35 49.9 1 = ° = ° A B 35 130.1 2 Now, use the Law of Sines to find the third side c. = = ° = ° ° = ° = ° ° ≈ = ° ° ≈ A a C c A a C c c c c c sin sin sin sin sin35 6 sin95.1 sin35 6 sin14.9 6 sin95.1 sin35 10.42 6 sin14.9 sin35 2.69 1 1 2 2 1 2 1 2 The two solved triangles are illustrated in Figure 30(b). Now Work PROBLEMS 27 AND 33 3 Solve Applied Problems Finding the Height of a Mountain To measure the height of a mountain, a surveyor takes two sightings of the peak at a distance 900 meters apart on a direct line to the mountain.* See Figure 31(a) on the next page. The first observation results in an angle of elevation of ° 47 , and the second results in an angle of elevation of ° 35 . If the transit is 2 meters high, what is the height h of the mountain? EXAMPLE 6 *For simplicity, assume that these sightings are at the same level. (continued)
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